“The main error to be avoided is trying to attack the problem head-on” – Alain Connes
“Face problems with a minimum of blind calculation, a maximum of seeing thought” – Hermann Minkowski
Prove_C1_book_wrong_on_differentiation (a useful exercise, I think, on several levels: learning what differentiation is; learning how to define things; curve-sketching; learning to distrust textbooks).
A lot of school maths is about being shown a procedure, practising it, and becoming fluent with it. For example, you’re shown how to solve quadratic equations. You know you’re good at it when you can solve any new quadratic equation accurately and fast.
That sort of fluency is a necessary basis for further progress in maths. But the further progress is something else again. It demands creativity and imagination. You deal with problems which come with few clues, and no obvious clues, about what method to use; or where the “obvious” method won’t work and you have to think up a non-obvious one.
You deal with problems where, normally and routinely, your first reaction is just to be baffled. MAT and STEP problems are like that; so are many “real-life” maths problems.
First hint: don’t give up. Expect to make some false starts, or sometimes, for a while, not to be able to think of any starts at all, even false ones. It may take you some time to imagine how to deal with the problem. That’s all right. Think of as many different ways of approaching the problem as you can. Try each one out. Most won’t work. Maybe one will.
When asked how he got his results, Carl Friedrich Gauss, reckoned the cleverest mathematician ever, replied: “By trying things out systematically”.
We won’t be as imaginative about different things to try out as Gauss was, or as quick and fluent in trying them out. But the same motto is valid for all of us.
Another motto for us all comes from the modern-day French mathematician Alain Connes: “The main error to be avoided is trying to attack the problem head-on”.
1. Draw a picture showing the information you have or want. If you can’t see how to draw a picture, construct a table or some other compact representation of the problem, or at least a table of what you know or what you want.
2. In your picture, have you included the relevant information? Stripped away the irrelevant? Simplified the relevant as much as you can?
3. In many real-life cases, though not in MAT or STEP or maths competition problems, there may be information which you need to get an answer, but don’t yet have. If so, how can you get it, or at least a guess at it?
4. Try looking at the problem in different ways, i.e. not just “head-on”. For example, you prove that the angle at the centre of a circle is twice the angle at the circumference essentially by switching from seeing the circle as a round thing to seeing it as a lot of isosceles triangles with the same vertex. A lot of problems in 2D maths can be seen as problems in geometry, in algebra via coordinate geometry, in vectors, in complex numbers, or in matrix algebra: switching from one way of looking to another (and then maybe back again) often helps. We can link measure and number theory by seeing the volume of a square pyramid (or a tetrahedron) as more or less the same as the sum of a series of square (or triangular) numbers. And so on.
5. Review what mathematical rules and theories you know which may help. Write down a list.
6. Try a simpler case, or a similar problem with small numbers. Example: the problem is to work out in your head whether 910 is bigger than 109.
Look at a simplified version of the same problem. Observe that 12 is less than 21. 23 is still slightly less than 32, but 34 is bigger than 43. The trend suggests that 910 is bigger than 109.
It also suggests more: that nn+1 is bigger than (n+1)n for all n bigger than 2.
Then do the algebra, dividing [nn+1/(n+1)n] by [(n-1)n/nn-1] to get [(n2)n]/[(n2-1)n]. That confirms that nn+1 does indeed increase faster than (n+1)n.
7. Refine and generalise to get the maximum out of a problem. This isn’t important in MAT or STEP or maths competitions, but it is in real life.
The problem of working out in your head whether 910 is bigger than 109 illustrates that, too. The method above not only answers the problem you were asked, but tells us that nn+1 is bigger than (n+1)n for all n>2. With a small step further, it shows us that (n+1)n/nn+1 approaches e/n as n becomes larger.
8. Try to break down the problem into steps by finding an intermediate calculation or result which will not get you to the answer, but will get you nearer, or give you an idea. For example, if a right-angled triangle has whole-number sides, the product of those three whole numbers is divisible by 60. (Examples: 3, 4, 5; 5, 12, 13; 8, 15, 17; etc.). How to prove that? You will probably be baffled at first. But you may be able to see quickly why the product must be divisible by 2, and build on the method used there to prove divisibility by 60.
9. Make a guess at the answer. (If the problem is to prove a particular claim, then you already know the “answer”). See if you can work backwards from the answer (or guess), as well as forwards from the question, to find an intermediate step which will help you.
10. If you find yourself getting lost in complicated calculations, pause, backtrack, check. If that doesn’t clear things up, rewrite all your working neatly, clearly, with lots of space. Check it with simple cases or small numbers.
11. Use symmetry. Take a step back from the problem and see what symmetry there is in it which you can use to narrow down the solution.
12. Once you have solved the problem, look to see if there is a simpler, neater solution, or if you can generalise your method to bigger or other problems. Think of what you’ve learned from that problem which may help you with other problems.
Some further comments:
Tony Gardiner stresses, and I think he’s right despite his grumpy-old-man manner, that ready recall and fluency with methods, theorems, and algorithms is a necessary base for all the above. You can no more teach problem-solving in abstraction from that drill than you can teach someone to find their way round a city without them getting to know and recognise and recall the landmarks and the main roads.
Picture-drawing is explicitly recommended by Terry Tao – “a picture is worth a thousand equations” – and in his little book on problem-solving it is his first step in almost every problem.
The “working backwards” is useful even with Edexcel problems in proof by induction, for example.
Trying out simpler cases or small numbers is also a repeated theme with Terry Tao, and explicitly advocated by many others including Dusa McDuff.
In Terry Tao’s worked problems in his little book, breaking the problem down into intermediate steps is a constant theme.
Kevin Houston suggests the discipline of considering, with every claim we prove, whether the converse is also true. Sometimes it obviously is true, sometimes it obviously isn’t. Fairly often it’s an interesting question. Example: if c2 = a2 + b2 for the sides of a triangle a, b, c, does it follow that the triangle is right-angled?
Not only Edexcel problems, but also maths-competition problems, usually give all the relevant information clearly and with very little irrelevant addition. Real maths is unlike that both at research level and at everyday “street maths” level. We have to learn how to separate out relevant and irrelevant information.
G H Hardy wanted to abolish the Maths Tripos exams at Cambridge – in fact he won only partial reforms, like abolishing the custom of announcing the order of the candidates, Senior Wrangler and so on – because they geared the best students to a particular sort of cunning suitable for maths-competition-type problems. They did that as a result of previous reforms moving the exams away from Edexcel-type problems. Both types are specialised, and leave vast important areas neglected.
Fermi problems, in other words problems where you seek (at least an approximate) answer without having all the information you need, develop mathematical skills neglected by the usual process in school maths of being shown a procedure and then practising it until you become fluent. See
From my experience with year 10 students in Queensland, Fermi problems are accessible and useful for the whole range of students.
One of the bad things about Edexcel problems is that they all have exact answers. (Look at the back of the book, or the mark schemes). Much everyday, and much academic and industrial, use of maths is instead about good approximate answers.
I’d suggest three rules for Fermi problems. Have a good sense of number and mental arithmetic. (I doubt Edexcel-maths helps a lot here, for example to appreciate straight off that the 90% of the numbers between 1 and 999,999 have six figures). Have a good range of general information about rough magnitudes. And break down problems: get an estimate of something for which you have no idea straight off by breaking it down into getting estimates of intermediate quantities.
Another bad thing about Edexcel problems is that they are all “straight ahead” problems. Anyone who has any chance at all of passing the exam can tell straight off what module from the course each problem is based. The road to follow is signposted. You may have difficulty remembering clearly the road map you learned, or the problem may involve a lengthy and complicated journey down that road, with a lot of working, but there is no puzzle about which road to take. Some Edexcel questions go so far as to state in the question the method that should be used, and to penalise in the mark scheme (with zero marks) other valid methods.
Alain Connes: “The main error to be avoided is trying to attack the problem head-on”. Edexcel-maths systematically teaches students to make a habit of that “main error”.
In Queensland all maths exams, even for students doing the minimum of maths to get by, must compulsorily include a number of two-step questions, questions in which the student has to change track halfway through and shift from one method to another. There is a significant minority of students who do better with the two-step questions than with the more technically-complicated one-step questions. In other words, two-step questions do not have to be reserved for the students who are best at or keenest on maths. Also, pretty much all Fermi problems have some multi-step character, and they are much more typical of everyday maths than are Edexcel problems.